What do people mean when they say (as when cautioning against the validity of a scientific study, for example) that "correlation does not imply causation"? Isn't causation just perfect correlation? And if so, doesn't that mean that the caveat in question does not concern causal claims per se, but inductive claims more generally?

Suppose there is a single type of event A that always causes two other events, B and C. Suppose, moreover, that, whenever B occurs, it is always caused by A, and similarly for C. Then B will be perfectly correlated with C, but by hypothesis is not caused by it. This is typically what people worry about, though of course the correlation is rarely perfect, in fact. For example, people with AIDS almost always have HIV. There's a very strong correlation. We also know now that HIV does cause AIDS. But the correlation by itself does not show that HIV causes AIDS. Maybe having HIV is actually another effect of whatever it is that actually causes AIDS. Your immune system is depressed, and so you have this virus that other people do not have. Not actually true, but it might have been true. But perhaps what is bothering you is a slightly different thought: This kind of strict correlation can't just be accidental, can it? Surely something must explain it! But, as I have said,...

On a Philosophy Bites podcast I heard Daniel Dennett mention the following thought experiment, which he attributed to Galileo. Suppose that heavier objects fall faster than light ones. Take two objects, A and B, where A is lighter than B. Connect A and B with a string and drop them. Since A is lighter than B, A will act as a drag on B, and B will fall more slowly than it would have alone. Yet since A and B are jointly heavier than B, and heavier objects fall faster than light ones, B will fall faster than it would have alone. We have a contradiction. Therefore, heavier objects do not fall faster than light ones. I thought that this was really marvelous and also very surprising. I had been under the impression that one could not arrive at ostensibly substantive empirical claims like the one in question just by considering thought experiments. I was hoping that one of the panelists could explain exactly how Galileo's thought experiment works here.

One way of thinking of this kind of thing is that it uncovers a subtle contradiction between views we already hold. But it doesn't, by itself, prove anything positive. It's conceivable that one could deny, for example, that A and B are "connected" in the right way for them to fall faster together. The question how thought experiments work in science is an important one, and philosophers and historians of science have written on it extensively. One good place to start might be Thomas Kuhn's paper, "A Function for Thought Experiments", which discusses this one, and also Einstein's famous thought experiment involving the moving train. There's a nice You Tube version of that one here. Check out the SEP article on thought experiments , as well.

Many people believe in the concept of a "soulmate". Do I need to share everything with my partner? Should my partner always be my best and closest friend?

I'm wary of these sorts of comparatives: best, closest. We all have many relationships that mean a great deal to us, and we do not need to make sure that one of them is "best", "closest". Indeed, if there is anything I've learned about relationships, it is how destructive those sorts of expectations can be. One certainly should not feel that if some interest one has isn't shared by one's partner, then it isn't worthwhile, or has to be sacrificed to the relationship, or what have you. That kind of thing starts to sound to me kind of clingy and possessive. There may be some counterexamples, but in most healthy couples I know, both partners have interests that are not shared (or not really embraced) by the other partner, and their freedom to explore and nurture those interests with other friends or family is part of what keeps them growing, both individually and as a couple.

How does one determine which side in an argument must shoulder the burden of proof?

The other guy has the burden of proof. And yes, I'm serious. It's that bad. But, to elaborate a little bit, I despise burden of proof type arguments. I do not know of any reasonable way of telling who "ought" to have the burden of proof, and I'm not sure I understand what is supposed to follow from someone's having it. People often end arguments saying something like, "Since they have the burden of proof and haven't met it, it is reasonable for us to believe my view". But this seems to me an odd way of thinking about philosophy. I mean, I do hope that some of the philosophical views I hold will have some influence and help us understand certain sorts of things better than we do. But whether any of my views might actually be true I very much doubt. And the fact that the other guy hasn't been able to knock my view down doesn't seem like good reason to believe it, even if my view is more common-sensical than his (a common test). Philosophy seems to me to be much more a hunt for understanding than...

Dear philosophers, I really appreciate your website, which I just discovered! I'd like to make one comment regarding the recent questions about infinite sets on March 7 and March 14. In your responses (Allen Stairs and Richard Heck on March 14), you write that you do not know of any professional mathematicians who deny the existence of infinite sets. However, such mathematicians do indeed exist (although marginally). They are sometimes referred to as "ultrafinitists". One well-known living proponent of this view is Princeton mathematician Edward Nelson, also see http://en.wikipedia.org/wiki/Edward_Nelson and http://en.wikipedia.org/wiki/Ultrafinitism Specifically, one argument an ultrafinitist might use is that formal proofs are finite. Thus, although we might use the concept of infinite sets in our reasoning, there is no need to assume that infinite sets actually exist, because any mathematical statement could be preceded by the phrase "There is a finite proof of the statement that ..." I hope this...

What I said was: "It's important to distinguish two different issues: (i) whether there are infinitely many natural numbers; (ii) whether there are mathematical objects that are themselves infinite. And it is possible to accept that there are infinitely many natural numbers without accepting that there is a set of all of them or, more generally, that there are any objects that are, in their own right, infinite. And there are respected mathematicians who hold this kind of view, though they are definitely a minority." So I was not saying that no professional mathematicians would deny the existence of infinite sets. Indeed, Nelson was very much the sort of person I had in mind. He may well be an example of a mathematician who does not think that there is a largest natural number, but who does not think that there are any infinite sets. But I'm not absolutely sure about this, due to the fact that much talk of infinite sets can be coded in the sorts of weak theories that Nelson would accept. I think...

Hi, I was hoping for some clarification from Professor Maitzen about his comments on infinite sets (on March 7). The fact that every natural number has a successor is only true for the natural numbers so far encountered (and imagined, I suppose). Granted, I can't conceive of how it could be that we couldn't just add 1 to any natural number to get another one, but that doesn't mean it's impossible. It seems quite strange, but there are some professional mathematicians who claim that the existence of a largest natural number (probably so large that we would never come close to dealing with it) is much less strange and problematic than many of the conclusions that result from the acceptance of infinities. If we want to define natural numbers such that each natural number by definition has a successor, then mathematical induction tells us there are infinitely many of them. But mathematical induction itself only proves things given certain mathematical definitions. Whether those definitions indeed...

I'm not familiar, either, with any working mathematicians who think there is a largest natural number or, more specifically, that there are only finitely many numbers. I do know of some work, by Graham Priest, that investigates finite models of arithmetical theories, but this is in the context of so-called paraconsistent logics. In Priest's theories, it is true that there is a greatest natural number, but it is also true that there isn't one! But that is probably not the kind of thing the questioner meant. Part of the reason mathematicians are happy with infinity is that infinity is very cheap. Consider, for example, ordered pairs. If you think (a) that, given any two objects, there is an ordered pair of them and (b) that there is an object that is not a pair, then it follows that there are infinitely many pairs. Or consider the English sentences. Not just the ones someone has uttered or written down, since there are ever so many English sentences no one happens to have uttered before (such, I am sure,...

If we assume that both computers and the human mind are merely physical, does it follow that a sufficiently advanced computer could do anything that a human brain could do?

No, because the mere physicality of the brain does not imply that the brain is any kind of computer. Maybe the brain is capable of various sorts of quantum computations that would allow it to perform tasks that no computer, even in principle, can perform. Who knows? Indeed, some people have argued that we can prove that the human mind can do things no computer can do, and these arguments do not imply that the mind is in any way non-physical. I think those arguments are no good myself, but they make this point anyway.

I read in one of my dad's linguistic books that some languages have exactly three basic color words: black, white, and red. I wondered if this meant that for the people who speak these languages, everything that is not black and white is called red (the sky is red, grass is red, etc.) -- or if they just don't have a word to describe anything that is not black, white, or red. If it is the latter, then how would they describe the color of the sky and grass? Noah L. Age 11

Hi, Noah, thanks for writing us with your question. I'm not sure which book you were reading, and I have never heard of such languages myself. To be honest, I kind of doubt there really are such languages. Have you ever heard about how Eskimos have lots of words for "snow"? Well, at least a lot of people think that's just wrong. It's a myth. In this case, I find it hard to imagine that the people speaking any language wouldn't find it useful to have words for more colors than the ones you mention. And if it's useful, then they will introduce such words. But let's suppose that there are languages like that and ask what we should say about them, if so. Both options you mention seem possible: that they have words for "black", "white", and "colored", and that they have words for "black", "white", and "red". In the latter case, then, as you say, they would have no word for the color of the sky. But they could still describe it, if they had a word meaning "same color". They could say the sky was the...

I hope this makes sense... I've always been curious about attempts to understand the way our minds work. To me, it seems paradoxical and in some ways even hopeless. I suspect that in order for the mind to understand or learn something new, the mind itself (or at least the way it works) needs to be more complex than what it it processing. In other words, the "size" of the new information cannot exceed the "capacity" of the mind itself in order to store it. An example of this would be the way computers work: Let's say I have a PC with an old operating system (Windows 2000) and I wish to run a software CD designed for a more advanced operating system (Windows 8). My old computer will most likely not recognize any of the information on that new CD, either because my old computer requires more free space (capacity of mind) or because the information stored on that CD requires a different kind of technology to decrypt (complexity of idea). Thus, you can use a computer to fully process programs (according to its...

I don't work in this sort of area myself, but this kind of view has been held. The position is known as mysterianism , and its main proponent is Colin McGinn . Considerations in the same ballpark also fuel the (in)famous arguments against mechanism due to John Lucas. What certainly does seem clear is that this kind of possibility can't be ruled out a priori. Surely there are some things human minds simply could not ever understand. That's true of all other creatures. Cats, for example, clearly do not have minds complex enough to understand calculus, let alone the nature of their own minds. We all have cognitive limitations. Perhaps we are in a similar position with respect to our minds. But it is not obvious either that our minds are limited in this particular way. The "self-reflective" aspect of understanding our own minds does not, by itself, show that we couldn't possibly do it. Your references to complexity and the like are suggestive, but there are many ways to measure of...

Is it wrong to desire sex with a woman when your primary interests are only physical and hence you might not even know or have spoken to the woman you desire sexually? Or does that only become problematic when a man expresses interest to that woman in a manner which is unsolicited and hence it becomes an unwanted and creepy sexual advance?

I don't see anything wrong about desiring to have sex with someone you don't know. I rather suspect this is a completely normal aspect of human sexual experience, and that it is simply a reflection of sexual attraction. Tons of people fantasize about sex with celebrities, for example, or some beautiful person they saw momentarily on a train, or what have you. Perhaps the word "desire" here is a bit unhelpful. Desires can be fleeting or life-long, momentary or sustained, deeply felt or like a twinge. I think it would definitely be strange, or even pathological, for someone to dedicate themselves to having sex with someone they'd merely seen, especially since they would have no reason whatsoever to think the desire mutual. That's not to say it would be creepy to try to take some steps to meet the person, but if one's only desire were for sex, then I think it is creepy again, since one has no reason to think the desire mutual. And sex isn't sex without mutuality, as this wonderful video makes...